R: Heteroskedasticity-Consistent Covariance Matrix Estimation

vcovHC {sandwich} | R Documentation

## Heteroskedasticity-Consistent Covariance Matrix Estimation

### Description

Heteroskedasticity-consistent estimation of the covariance matrix of the coefficient estimates in regression models.

### Usage

    vcovHC(x, ...)

    ## Default S3 method:
    vcovHC(x,
      type = c("HC3", "const", "HC", "HC0", "HC1", "HC2", "HC4", "HC4m", "HC5"),
      omega = NULL, sandwich = TRUE, ...)

    meatHC(x, type = , omega = NULL)

### Arguments

`x` | a fitted model object.
`type` | a character string specifying the estimation type. For details see below.
`omega` | a vector or a function depending on the arguments `residuals` (the working residuals of the model), `diaghat` (the diagonal of the corresponding hat matrix) and `df` (the residual degrees of freedom). For details see below.
`sandwich` | logical. Should the sandwich estimator be computed? If set to `FALSE` only the meat matrix is returned.
`...` | arguments passed to `sandwich`.

### Details

The function `meatHC` is the real work horse for estimating the meat of HC sandwich estimators – the default `vcovHC` method is a wrapper calling `sandwich` and `bread`. See Zeileis (2006) for more implementation details. The theoretical background, exemplified for the linear regression model, is described below and in Zeileis (2004). Analogous formulas are employed for other types of models.

When `type = "const"` constant variances are assumed and and `vcovHC` gives the usual estimate of the covariance matrix of the coefficient estimates:

sigma^2 (X'X)^{-1}

All other methods do not assume constant variances and are suitable in case of heteroskedasticity. `"HC"` (or equivalently `"HC0"`) gives White's estimator, the other estimators are refinements of this. They are all of form

(X'X)^{-1} X' Omega X (X'X)^{-1}

and differ in the choice of Omega. This is in all cases a diagonal matrix whose elements can be either supplied as a vector `omega` or as a a function `omega` of the residuals, the diagonal elements of the hat matrix and the residual degrees of freedom. For White's estimator

`omega <- function(residuals, diaghat, df) residuals^2`

Instead of specifying the diagonal `omega` or a function for estimating it, the `type` argument can be used to specify the HC0 to HC5 estimators. If `omega` is used, `type` is ignored.

Long \& Ervin (2000) conduct a simulation study of HC estimators (HC0 to HC3) in the linear regression model, recommending to use HC3 which is thus the default in `vcovHC`. Cribari-Neto (2004), Cribari-Neto, Souza, \& Vasconcellos (2007), and Cribari-Neto \& Da Silva (2011), respectively, suggest the HC4, HC5, and modified HC4m type estimators. All of them are tailored to take into account the effect of leverage points in the design matrix. For more details see the references.

### Value

A matrix containing the covariance matrix estimate.

### References

Cribari-Neto F. (2004), Asymptotic Inference under Heteroskedasticity of Unknown Form. Computational Statistics \& Data Analysis 45, 215–233.

Cribari-Neto F., Da Silva W.B. (2011), A New Heteroskedasticity-Consistent Covariance Matrix Estimator for the Linear Regression Model. Advances in Statistical Analysis, 95(2), 129–146.

Cribari-Neto F., Souza T.C., Vasconcellos, K.L.P. (2007), Inference under Heteroskedasticity and Leveraged Data. Communications in Statistics – Theory and Methods, 36, 1877–1888. Errata: 37, 3329–3330, 2008.

Long J. S., Ervin L. H. (2000), Using Heteroscedasticity Consistent Standard Errors in the Linear Regression Model. The American Statistician, 54, 217–224.

MacKinnon J. G., White H. (1985), Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties. Journal of Econometrics 29, 305–325.

White H. (1980), A Heteroskedasticity-Consistent Covariance Matrix and a Direct Test for Heteroskedasticity. Econometrica 48, 817–838.

Zeileis A (2004), Econometric Computing with HC and HAC Covariance Matrix Estimators. Journal of Statistical Software, 11(10), 1–17. URL http://www.jstatsoft.org/v11/i10/.

Zeileis A (2006), Object-Oriented Computation of Sandwich Estimators. Journal of Statistical Software, 16(9), 1–16. URL http://www.jstatsoft.org/v16/i09/.

### See Also

`lm`, `hccm`, `bptest`, `ncv.test`

### Examples

    ## generate linear regression relationship
    ## with homoskedastic variances
    x <- sin(1:100)
    y <- 1 + x + rnorm(100)
    ## model fit and HC3 covariance
    fm <- lm(y ~ x)
    vcovHC(fm)
    ## usual covariance matrix
    vcovHC(fm, type = "const")
    vcov(fm)

    sigma2 <- sum(residuals(lm(y ~ x))^2)/98
    sigma2 * solve(crossprod(cbind(1, x)))

[Package sandwich version 2.3-4 Index]